# Mathematics ## Please practice saying what you think. Feel free to disagree with what I say and prove a contradiction!The Riemann hypothesis is proved in Number, space and logic (NSL), volume II chapter 6, under informal review, using Dw exponential algebra developed by David Bohm and me.NSL volume I chapter 4, on adonions which are division algebras and novanions (these are all called zargonions), is used in the Physics section. Adonions beyond the octonions exist but are supposed not to.Games of the form (+, 0, -) exist throughout social systems, mathematics and physics. Other ordered game triples are (cooperative, control or competitive, uncooperative), (ethics, Kampf wall, murder), in solving polynomials (greater degree comparison, stable degree module, lesser degree polynomial) and in physics (J2 = -1 global interaction zargonion, J2 = 0 tribble, J2 = 1 quantum tharl algebra) aligned by 'overwhelming'. So with the Kampf wall down, murder with death overwhelms left ethics with a currency of need.For applications in engineering conceptors are meaning processors rather than syntax processors (computers).We replace Galois theory using dependent roots, ring automorphisms and varieties. The quintic is not solvable by 'killing central terms' but originally we had thought it is solvable in radicals by 'polynomial wheels', but other methods have to be used. It is also solvable using approximation methods.

I have finally after 54 years refuted Galois solvability theory by proving the sextic is solvable.
This has been my aim since the age of 15.
Most decisively of all, as will become evident, it generalises to all polynomial equations.
Every polynomial equation is solvable in radicals, and therefore every consistent algorithm.
The most alarming aspect is the control wall of the theory of modules, and thus homological algebra.
The theory is almost not there - a black hole. It is like the burning of the Library of Alexandria.

eBook: Weird glyph games and hyperintuition
eBook: Stupidity, confusion, distractions and bamboozlement

The Unmentionable

The revolution Number, space and logic (NSL) contains sixteen months work.
There will be an exposition and extension of homolgy, cohomology and homotopy theory in it.

Refs: Fred Diamond and Jerry Shurman, A first course in modular forms, Springer, 2005, which is strongly recommended,
Roger Carter, Finite Groups of Lie Type, J. Wiley, 1985,
J. H. Conway and N. J. Sloane, Sphere Packings, Lattices and Groups, Springer, 1998,
John H. Conway and Derek Smith, On Quaternions and Octonions, A. K. Peters, 2003,
J. P. May, A consise course in algebraic topology, U. Chicargo P., 1999,
Hershel Farkas and Irwin Kra, Riemann Surfaces (2nd edn.), Springer, 1992 and
M. E. Szabo (ed.) The collected papers of Gerhard Gentzen, North-Holland, 1969.

1A. Insight on Computable Superstructures

Chapter 1
1B. The Meaning of the Finite and the Infinite

We introduce a modified set theory. We extend the rules for the positive whole numbers to transfinite natural numbers, called
transnatural numbers. Some basic properties of the transnatural numbers are developed. We define a model for the real numbers using 'capital Zeta functions', deconstruct the uncountability assumption for the set of subsets of the natural numbers and introduce an algebra of transinfinite ordinals, called ladder algebra.

1.1 Introduction
1.2 Formal language
1.3 Modified Zermelo-Fraenkel set theory (mZFC)
1.4 The Peano axioms for the transnatural numbers, ??t
1.5 A model for the real numbers
1.6 The capital ? function
1.7 The Euclidean algorithm
1.8 Fermat's little theorem
1.9 Euler's totient formula
1.11 The inconsistency of the uncountability hypothesis for ??
1.12 Fields and zero algebras
1.13 Ordinal (ladder) arithmetic in ??t and ?t
1.14 Spiritissimos

Chapter 2
1C. The Meaning of Branched Spaces.

We discuss lattices. Removing a point from a real line divides it into two pieces. So we say removing a point from an n-branched space
divides it into n pieces. We introduce a model for a number descibing branched space topology, the Euler characteristic, as a general polynomial with integer coefficients.

2.1 Introduction
2.2 Vector spaces, scalar products and modules
2.3 Boxes and box scalar products
2.4 The probability logic of dependent and independent events
2.5 Posets and lattices
2.6 Multilattices and the Dedekind-MacNeille construction
2.7 Finished and unfinished sets
2.8 Twisted and untwisted logic
2.9 The Euler-Poincaré characteristic
2.10 The familiar square, cylinder, torus and cube
2.11 Branched spaces
2.12 Models for branched lines and areas
2.13 Models for multiobjects
2.14 Deformation retracts and orientation
2.15 Branched handles, crosscaps and surgery of branched spaces
2.16 Explosion boundaries
2.17 The general polynomial
2.18 Trees, amalgams and glyphs

Chapter 3
The meaning of suoperators and categories

1D. Groups Chapter 3. We describe basic group theory and prove the three isomorphism theorems. We introduce the Schur multiplier and look at the standard classification of simple
groups. We also prove the orbit-stabiliser theorem, Sylow's theorems and the Jordan-Hoelder theorem. We give a counter-example to the statement that normal groups form a modular
lattice. We describe the classification of Lie algebras.

3.1 Introduction
3.2 Notation
3.3 Intricate and hyperintricate numbers
3.4 The modular group
3.5 Polyticate numbers
3.6 The j-abelian property
3.7 Dw exponential axioms
3.8 Dw suoperator axioms
3.9 Crude suoperators
3.10 Zargon suoperators
3.11 Nonassociative representations of sunomials
3.12 Sunorms and branching
3.13 Subsunomials and singularities
3.14 Suderivatives
3.15 Suintegration
3.16 Interaction and descent of sunomials
3.17 The meaning and ideas of category theory
3.18 Functors
3.19 Universals
3.20 Posets, lattices and graphs

Chapter 4
Zargonions

1E. Zargonions Chapter 4. A sequence of algebraic structures begins with real numbers, complex numbers, the quaternions and then octonions. Adonions are a generalisation of
octonions, generating algebras with division of dimension 6k + 2 for whole numbers k. This is in violation of the result of J.F. Adams On the Nonexistence of Elements of Hopf Invariant
One
, which uses Steenrod squares, equivalent to using K theory. But K theory does not consider sporadic groups, and this is a gap in the proof. This survey collects together results on
division algebras, the novanions of Superexponential Algebra and their combinations, called zargonions. Additional to the chapter in Superexponential algebra, we show that the 10- and
26-novanions have both fermionic (odd number of twists) and bosonic (even number of twists) representations, and there is an enveloping novanion. We describe the classification of
vertex operator algebras.

4.1 Introduction
4.2 Division algebras
4.3 Zargon algebras
4.4 The quaternions
4.5 The nonassociative octonion division algebra
4.7 The classification of novanions
4.8 Eigenvalues
4.9 The 10-novanions
4.10 n-novanions
4.11 The search for other novanion algebras
4.12 The 64-novanions
4.13 The García classification problem
4.14 The generation and classification of zargon algebras
4.15 Zargon algebras of negative dimension
4.16 Zargon brackets
4.17 Discreteness and zargon rings
4.18 Tharlonions and tribbles
4.19 The origin of the discrepancy of our results with K theory

Chapter 5
The meaning of superstructure games

1F. The Meaning of Suoperators Chapter 5. Just as addition by repetition generates multiplication and multiplication by repetition generates exponentiation, so an nth superexponential
operation, called a suoperator, generates an (n + 1)th suoperator. We introduce a general algebra for suoperators in an extended form from that given in the eBook Superexponential
algebra
, including a discussion of tharl algebras and tribbles. We discuss generalisations obtained from the suoperators of plus, times and multiplication of differentiation (sudifferentiation),
integration (suintegration) and of the binomial theorem.

1G. The Meaning of Categories and Superstructures Chapter 6. We give a survey of category theory, including a discussion of choice, functors, universals, adjoints, limits, colimits,
comma categories, multilimits, the Wonderful Theorem and Kan extensions. We generalise to the nonassociative case called superstructure theory, considering toposes and sutoposes.

1H. The Meaning of Superstructure Games Chapter 7. We introduce a discussion of games in the context of superstructure theory.

1I. Algorithms Chapter 8. We describe Birkby's theorem, algorithms and hyperintuition.

1J. The Discovery of the Polynomial Wheel Chapter 9. This chapter needs considerable revision. Polynomial wheels are general techniques to solve polynomial equations of any degree by radicals, previously thought
impossible. We introduce polynomial wheel theory and comparison methods and survey similar results to Superexponential algebra volume II, demonstrating that the Galois solvability
model is incorrect for comparison methods which avoid killing central terms, show by general methods that solutions of polynomial equations of arbitrary degree can always be found.
All general arguments of this type can be re-expressed in category theory), provide a geometric realisation of the cubic, and look in detail at the solvability of quintic polynomials. We
review the solvability of the quintic as embedded in a quartic variety in squared variables, showing a link with elliptic curves. The general degree descent techniques are called polynomial
wheel methods and give for example solutions of the quintic by radicals.

1K. Consistent Problems are Solvable Chapter 10. The quintic gives a solution of the sextic. This is part of a general theorem that a solvable polynomial of odd degree n gives rise
to a solvable polynomial of degree n + 1. This process can be iterated indefinitely and is part of a general theorem that all consistent problems are solvable - unsolvable problems are
inconsistent. Solutions by comparison methods are further investigated. We generalise the discussion on elliptic curves of chapter 7.

5.1 Introduction
5.2 Making breakfast - ethical game theory at work
5.3 The meaning and ideas of superstructure theory
5.4 Toposes and sutoposes
5.5 Limits, colimits and choice
5.6 Comma categories and superstructures
5.7 Kan extensions
5.8 Pure simple games and the definition of overwhelming
5.9 Semantics in Kogito and Fizyk
5.10 Zargon games
5.11 Some evolutionary processes
5.12 Insight algebra
5.13 Hyperintuition

Chapter 6
Groups

6.1 Introduction
6.2 Basic ideas in group theory
6.3 Normal subgroups
6.4 The isomorphism theorems
6.5 The Schur multiplier
6.6 The standard classification of simple groups
6.7 The orbit-stabiliser theorem
6.8 Sylow's theorems
6.9 Composition series
6.10 Normal subgroups not forming a modular lattice
6.11 Continuous groups
6.12 Dynkin diagrams
6.13 Vertex operator algebras

Chapter 7
The discovery of the polynomial wheel

7.1 Introduction
7.2 A history of Galois solvability and Galois representation ideas
7.3 Deconstructions of Galois solvability theory
7.4 Ring automorphisms
7.5 Dependency theory restrictions
7.6 A geometric realisation of the cubic
7.7 Birkby's theorem
7.8 Abstract underpinnings of polynomial wheels. First example.
7.9. Second example. An additive and multiplicative game.
7.10 Discriminants, the Sylvester determinant and Bring-Jerrard form
7.11 The comparison method for the quintic and its elliptic curve
7.12 The solution of the sextic given the solution of the quintic
7.13 A polynomial wheel method for solving the quintic polynomial in radicals
7.14 The solution of a general polynomial in radicals
7.15 Solutions for Gaussian integer degrees

VOLUME II:
Finite and Infinite Number Theory

2B. Number Theory Chapter 1 is half constructed. We look at ladder algebra and its transcendental extensions. The chapter has a section on Gauss, Ramanujan and Kloosterman sums.

2C. Class Field Theory Chapter 2 is not ready. We look at class field theory.

2D. Theta Functions Chapter 3 is not ready. We look at the work of Jacobi and Cayley on theta functions.

2E. The Consistency of Analysis Chapter 4 is not ready. We develop a Gentzen-type proof of the consistency of analysis from our theory of trees and our transcendental results. The
theory is extended to cover colour logics, also those of intuitionalistic type.

2F. Fermat's Last Theorem Chapter 5 is half complete. We prove Fermat's Last Theorem using elementary methods and give a more sophisticated treatment using the modularity theorem.

2G. Zeta Functions Chapter 6. The chapter proves the classical theory of the Riemann hypothesis using invariance under imaginary Dw exponential algebras and will prove the general
Riemann hypothesis by extending w.

2H. Eisenstein Series Chapter 7 is not ready.

2I. The Goldbach Conjecture Chapter 8 is not ready. The chapter proves the weak Goldbach theorem using the work of Harald Helfgott. The general Riemann hypothesis implies a new
and shorter technique for proving the weak Goldbach conjecture.

Chapter 1
Number theory

1.1 Introduction
1.2 Finite arithmetic
1.4 The Goodwin bijection

Chapter 2
Class field theory

2.1 Introduction
2.2 Algebraic congruence arithmetic
2.4 Gauss, Ramanujan and Kloosterman sums,br> 2.5 The Rogers-Ramanujan identities

Chapter 3
Theta functions

3.1 Introduction

Chapter 4
The consistency of analysis

4.1 Introduction,br> 4.2 V = L
4.3 Forcing
4.4 Gentzen's consistency theorem for arithmetic
4.4 A Gentzen-type proof of the consistency of analysis

Chapter 5
Fermat's last theorem

5.1 Introduction
5.2 Fermat's last theorem and elementary methods

Chapter 6
Global zeta functions

6.1 Introduction
6.2 ?-functions
6.3 The basic problem
6.4 Complex Dw exponential algebras
6.5 The proof of the Riemann hypothesis
6.6 L-series

Chapter 7
Eisenstein series

7.1 Introduction

Chapter 8
The Goldbach conjecture

8.1 Introduction
8.2 The Helfgott proof of the weak Goldbach theorem
8.3 The proof via the Riemann hypothesis

VOLUME III:
Algebra and Logic Combining Trees

4B. Trees and Amalgams Chapter 1 is not ready. We define trees and amalgams.

4C. Branched Spaces and Explosions Chapter 2 is not ready. We describe branching which is a general feature of superexponential algebras. The branching can be infinite in a set called
an explosion. If the algebraic structure is removed, the topology remains.

4D. Surgery Chapter 3 is not ready. We obtain from the branched Euler characteristic in multiplicative form, a series of operations called surgery to enhance the polynomial to a desired
additive form. This algebraic series of operations has a geometric realisation.

4F. Sequent Calculus and Colour Logic Chapter 4 is not ready. We look at multivalued logics, called colour logics, develop Gentzen's sequent calculus for logical deduction and extend
the discussion of sequents to modal and colour logic.

4G. Probability Logic, Distributions and Zargonion Fractals Chapter 5 is not ready. The chapter looks at probability logics from the point of view of distributions, and how to weight
polynomial curve fitting so that the resulting curve has maximum significance. It revisits colour logic in the probability context as a multidimensional logic, extends this to zargonions, and
then incorporates dimension in terms of zargonions with real coefficients as fractal structures.

4E. Optimisation and Complexity Chapter 6 is not ready. We discuss P/NP problems in the context of Birkby's theorem.

4H. Glyphs Chapter 7 is not ready.

Chapter 1
Trees and amalgams

1.1 Introduction
1.2 Undirected graphs
1.3 Cubic graphs
1.4 Frucht's theorem
1.5 Directed graphs
1.6 Trees
1.7 Free groups
1.8 The Nielsen-Schreier theorem
1.9 Branched retracts
1.10 Amalgams

Chapter 2
Branched spaces and explosions

2.1 Introduction
2.2 The suoperational model
2.3 Metrical and branched sunomial equivalence
2.4 Topological branching and explosions

Chapter 3
Surgery

3.1 Introduction

3.2 Surgery on unbranched spaces
3.3 The idea of surgery on branched spaces
3.5 Multiplicative maps
3.6 Superstructural maps

Chapter 4
Sequent calculus and colour logic

4.1 Introduction
4.2 Introduction to colour sets and other set theories
4.3 Suvarieties and colour logic
4.4 Classical sequent calculus and Gentzen normal form
4.5 A colour model for modal logics
4.6 Colour sequents

Chapter 5
Probability logic, distributions and Zargon fractals

5.1 Introduction
5.2 The propositional probability mapping to predicate logic
5.3 Distributions
5.4 Polynomial fit
5.5 Weighted distributions
5.6 The maximum significance theorem
5.7 Colour probability
5.8 Zargon probability logic
5.9 Zargon dimension and fractals

Chapter 6
Optimisation and complexity

6.1 Introduction
6.2 Linear programming
6.3 Hamiltonian circuits
6.4 Minimum graph colouring
6.5 Petri nets
6.6 Glyph optimisation
6.7 P/NP problems
6.8 Polynomial wheel complexity
6.9 Sunomial measures of complexity
6.10 Zargon Voronoi diagrams

Chapter 7
A general theory of glyphs

7.1 Introduction
7.2 The Wonderful theorem for colour logics
7.3 Algebraic structures associated to a glyph
7.3 Boundary flow and no boundary flow
7.4 Hyperintuition
7.4 Weird glyphs - no specified global algebraic structure

VOLUME IV:
Varieties and Topology

3B. Zargon Varieties Chapter 1 is not ready. We describe a generalisation of the theory of elliptic curves.

3C. Modular Forms - Representation Theory Chapter 2 is not ready. We include work on modular forms, used in the modularity theorem.

3D. Conformal and Nonconformal Analysis Chapter 3 is half ready. We include work on the hyperintricate Cauchy-Riemann equations and the Cauchy integral formula.

3E. Conformal and Nonconformal Suanalysis Chapter 4 is not ready. This develops suoperator analogues of the previous chapter.

3F. Riemann Roch Chapter 5 is not complete. This chapter includes work on the Riemann-Roch theorem, the Gauss-Bonnet theorem and stereographic projection.

3G. Homology Superstructures Chapter 6 is not ready. We describe our replacement of homology in the superexponential context.

3H. Cohomology Superstructures Chapter 7 is not ready. The chapter describes etale and motivic cohomology in the superexponential context.

3I. Homotopy Superstructures Chapter 8 is not ready. We describe homotopy, a theory of paths, in the superexponential context. Epicycle knots are discussed.

3J. Local Zeta Functions Chapter 9 is not complete. We look at the Weil conjectures. Implications for the function field case are given, where transl-adic and other results are applied.

Chapter 1
Matrix and zargon varieties

1.1 Introduction
1.2 Elliptic curves
1.3 Rational points on derived elliptic curves
1.4 Eigenvalues of matrix polynomials
1.5 Homogenous matrix varieties in two variables.
1.6 Extension maps and contraction maps.
1.7 Zargon suboxes
1.8 Zargon varieties
1.9 Zargon twisted manifolds

Chapter 2
Modular forms - representation theory

2.1 Introduction
2.2 Galois representation theory
2.3 Quadratic and general totient reciprocity
2.4 Hyperintricate and hyperpolyticate numbers
2.5 Eigenvalues and hyperactual numbers are J-abelian
2.6 Two dimensional lattices and Heegner numbers
2.7 Two dimensional lattices and abelian groups
2.8 Two dimensional lattices, topology and divisors
2.9 Two dimensional lattices and elliptic curves
2.10 Two dimensional lattices and theta functions

Chapter 3
Conformal and nonconformal analysis

3.1 Introduction
3.2 Fourier series
3.3 Convolutions
3.4 Good kernels, the Cesŕro mean and Fejér's theorem
3.5 The Maurer-Cartan equation
3.6 The intricate analytic Cauchy-Riemann equations
3.7 J-diffeomorphisms and non-fixed J
3.8 Intricate components and relative orientation
3.9 The intricate Cauchy-Riemann equations are nonconformal
3.10 The zargon analytic Cauchy-Riemann equations
3.11 The complex Cauchy integral formula
3.12 The zargon Cauchy integral formula

3.13 Picard's theorems
3.14 Singularities of mixed twisted and untwisted structures
3.15 Nonconformal structures are mixed and singular

Chapter 4
Conformal and nonconformal suanalysis

4.1 Introduction
4.2 Dw superstructures
4.3 Stepping down and stepping up
4.4 Sudifferentiation
4.5 Suintegration

Chapter 5
The Riemann-Roch theorem

5.1 Introduction
5.2 The Riemann-Roch theorem
5.6 Diffeomorphisms
5.7 Bézout's theorem
5.8 The Gauss-Bonnet theorem
5.9 The Atiyah-Singer index theorem
5.10 Embeddings and projections
5.11 The Lefschetz fixed point theorem

Chapter 6
Homology superstructures

6.1 Introduction
6.2 Orientation and antiorientation
6.3 Singular homology and cohomology
6.4 Ext and Tor

Chapter 7
Cohomology superstructures

7.1 Introduction
7.2 Our view on K theory
7.3 Generalised Steenrod squares
7.4 Characteristic classes
7.5 Étale cohomology
7.6 Crystalline cohomology
7.7 Some comments on motivic cohomology

Chapter 8
Homotopy superstructures

8.1 Introduction
8.2 Rotations, reflections and gluing
8.3 Circles, ellipses and Dw superstructures
8.4 Zargon rotations as knots and twistings
8.5 Epicycle knots
8.6 Quillen homotopy
8.7 Zargonions and Bott periodicity
8.8 Mixed structures and the Poincaré conjecture

Chapter 9
Local zeta functions

9.1 Introduction
9.2 Background and history of the Weil conjectures
9.3 The statement of the Weil conjectures
9.4 The projective line
9.5 Projective space
9.6 Elliptic curves
9.7 Weil cohomolgy
9.8 Grothendieck's proof of three of the four conjectures
9.9 Deligne's first proof of the Riemann hypothesis conjecturbr>e 9.10 Deligne's second proof

VOLUME V:
Sphere Configurations and Simple Groups

5B. Zargon Lattices Chapter 1. We discuss lattices from the zargonion point of view. We study the classification of groups, called zargon groups, including simple groups, arising
from the algebra of the zargonions, and investigate the relation of these new results with those obtained by using Dynkin diagrams. We obtain Lie algebras beyond those obtained from
the octonions, with more elements than E8. We obtain some simple groups with similar but different size to the monster, and an infinite sequence of such simple groups. We discuss
implications for quantum mechanics.

5C. Lattice Theta Functions Chapter 2 is not complete.

5D. Kissing Numbers Chapter 3 is not complete.

5E. Hamming and Golay codes Chapter 4 is not complete.

5F. The Novanion 9-1 Lorentzian Lattice Chapter 5 is not complete.

5G. The Adonion 13-1 Lorentzian Lattice Chapter 6 is not complete.

5H. The Novanion and Adonion 25-1 Lorentzian Lattices Chapter 7 is not complete.

5I. Moonshine Chapter 8 is not complete.

5J. The Adonion 31-1 Lorentzian Lattice Chapter 9 is not complete.

5K. Enveloping Zargon Moonshine Chapter 10 is not complete.

5L. References and Index Not ready.

Chapter 1
Zargon lattices

1.2 Introduction and history
1.3 Diagrams
1.4 Tharlonions and quantum mechanics
1.5 For group theory, polynomial wheels are not relevant

Chapter 2
Sphere packings and kissing numbers

2.1 Introduction
2.2 Sphere packings and kissing numbers

Chapter 3
Hamming and Golay codes

3.1 Introduction
3.2 Hamming codes
3.3 The Golay code

Chapter 4
The adonion ?7,1 and novanion ?9,1 Lorentzian lattices

4.1 Introduction
4.2 The adonion ?7,1 Lorentzian lattice
4.2 The novanion ?9,1 Lorentzian lattice
4.3 The E8 lattice

Chapter 5

5.1 Introduction
5.2 The adonion 13,1 Lorentzian lattice
5.3 The Coxeter-Todd lattice

Chapter 6
The novanion and adonion 25,1 Lorentzian lattices

6.1 Introduction
6.2 The novanion and adonion ?25,1 Lorentzian lattices
6.3 The Leech lattice

Chapter 7
Moonshine

7.1 Introduction
7.2 Moonshine

Chapter 8

8.1 Introduction
8.2 The adonion ?31,1 Lorentzian lattice